Island of adventure: Discover epic mountains, vibrant cities and exotic beaches on a tour of Taiwan

At the end of a surprising week-long journey down the spine of Taiwan, I'm sitting on beautiful Baisha Beach, in the protected Kenting National Park, near the island's southern tip.

The golden sun sinking towards the horizon and the ocean lapping against my toes look and feel real enough, and the breeze flicking around the palm trees sounds exactly as it should.

But in diplomatic terms, this place doesn't exist. For Taiwan is an anomaly after the two sides in the Cold War packed up and left. Only 23 countries, mainly from the South Pacific and Africa, acknowledge this island as being a legitimate nation. Britain isn't one of them. Like almost everyone else, we recognise the communist government in Beijing as being the winners in the war for China, instead.

In 1949, General Chiang Kai-shek's defeated Nationalist army fled from Mao's Communists and took up residence in Taiwan. Since then, the Chinese civil war has never properly ended.

But although China and Taiwan point thousands of missiles at each other across the Taiwan Strait, there's not much point in worrying about hostilities breaking out again, despite occasional bouts of sabre-rattling.

A friendly Taiwanese businessman who chats frankly on the beach tells me: 'China wants Taiwan's economy. There will never be war.'

But he also expresses another view I hear time and again throughout the week: 'I'm just glad I live on this side,' he says. 'I go to China regularly and it's not good. Too many people chasing too few resources. You have to fight there. here, it's totally different.' he's got a point. Friendly, safe, welcoming, democratic, free and devoid of petty crime, 50,000 Britons go there every year, making us the number one European tourists in Taiwan. And the island is set to welcome even more.

Coastal gem: A woman practises yoga on Baisha beach in the protected Kenting National Park

My Taiwanese journey took in the capital, Taipei, a thoroughly middle-class city where ancient temples and steam-filled dim sum cafes squat cheek-by-jowl with cyber-age electronics shops and monorails.

For years, Taipei was Asia's ugly duckling, but times have changed. And the cherry on the cake is Taipei 101, the world's tallest building, until it was surpassed by Dubai's Burj Khalifa.

Taipei 101, an imposing office block, is supposed to look like a bamboo stalk, but closer up it resembles Chinese takeaway containers stacked on top of each other.

It has the world's fastest lifts that shoot you up to the 89th floor observatory at 37mph. Once out on the roof, you can marvel at how Taipei has transformed itself into an economic powerhouse, with apartment towers and glass offices gleaming in the sunlight.

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The following day, I take the new high-speed train south. A facsimile of the legendary Japanese bullet train, this spotless, sleek engineering marvel darts all the way down Taiwan on top of an elevated viaduct in just an hour and three-quarters.

I get off in the nondescript town of Chiayi, Taiwan's equivalent of Swindon. No need to visit there normally, but, today, 100,000 people are thronging the streets for the annual lantern festival. (The next is from February 16 to March 10, 2011.)

There are traditional orange paper lanterns and novelty lanterns made by children and prisoners, but the centrepiece looks like something from a Bond villain's lair. In the town square, a 30ft plastic cat sits on a turntable. After some speeches from dignitaries, the crowd is silenced and thousands of phone cameras point at the cat - which I later find out is a tiger to celebrate the Chinese Year of the Tiger. Tension mounts.

Suddenly, the tiger is lit up with hundreds of lights; it spins round and fireworks fly out from every direction accompanied by rousing music. It's like some mad cult.

Natural inspiration: The Taipei 101 building is supposed to resemble a bamboo stalk

Then I travel by bus further south for a gentler few days in old colonial Taiwan. The first port of call is Tainan, where the Dutch established a toehold on the island from 1624 to 1662. The beautifully preserved Fort Zeelandia, which was the first Dutch settlement on the island, is a stone castle.

A few miles away is Fort Provintia. Razed by the Chinese Qing invaders, it is now topped with Oriental rather than European defensive buildings. Both offer a fascinating glimpse into how European powers meddled on the island.

In Kaohsiung the following day, I am surprised to find it wasn't just the Dutch who were meddling in Taiwan - the British did, too. The handsome brick-built, former British consular residence at Takao, on high escarpments overlooking Kaohsiung on one side and the sea on the other, was the opulent base from which our merchants tried to do business with Taiwan and China in the 1860s and 1870s.

With their historic colonial past and high-tech manufacturing present, Tainan and Kaohsiung are like fascinating mini Hong Kongs. Before I leave Taiwan, I want to shift down a gear, and that's what brings me all the way to Kenting National Park.

I rent a bike at the Yoho Bike hotel, a cyclist-friendly beachside bolthole where you can park your bike in your room.

I cycle along the pristine coast road on a Taiwanese-built Giant mountain bike reflecting on the island: the mix of high-tech and history, Chinese and Japanese culture, and the sunny disposition of the Taiwanese despite the simmering threat from the National People's Liberation Army.

And yet the biggest surprise of all is that on this small island there are epic mountains, vibrant cities and a coastline that wouldn't look out of place in South-East Asian honeytraps.

Taiwan might not technically exist, but as I sit on its best beach at dusk, drinking an ice cold local beer, its pull feels wonderfully real.